Why study curved space?

One of the things Astronomers get flak for is presuming that the universe is curved (or more precisely, their model permits the universe to curve). People think astronomers are losing touch by using these horribly abstract mathematical ideas that obviously can have no connection to physical reality!

But people don't know the reasons!

People think that astronomers are making all these additional assumptions about how the universe must look... but what they don't realize is that the exact opposite is true; astronomers are removing assumptions.

Something that people don't realize is that the "obviously correct" idea of a Euclidean space is chock full with a lot of assumptions; space looks the same on small and large scales (I mean the geometry, not necessarily the content)... squares exist... perfectly parallel lines exist... spheres have insides and outsides... lines intersect in at most one point...

All in all, when keeping with the spirit of Euclid, Hilbert axiomized euclidean geometry using fifteen postulates... and that's just for the geometry of the plane! People may remember that Euclid only had 5 postulates, but what they don't realize he also included some obvious rules which amount to additional assumptions, and Euclid was even incomplete! For example from Euclid's postulates you cannot prove that a line segment with one endpoint inside of a circle and one endpoint outside of a circle actually intersects the circle!

If we take a few steps back, we realize we're making a lot of assumptions, but then again, space certainly looks euclidean, doesn't it? But we realize that we're only looking at the space near us. It would make sense, then, presuming the universe is Euclidean on small scales is a much more reasonable law than presuming the entire universe is Euclidean!

And that's just what astronomers do; instead of presuming the entire universe is Euclidean, they presume it simply looks Euclidean on small scales (and may or may not look Euclidean globally). The mathematician Riemann studied geometries that look Euclidean on small scales, so today we call such things "Riemann manifolds". (they're a special type of "differentiable manifold")

However, as often happens in mathematics, the above reasoning is not why Riemann was studying his manifolds! He was active in a subject called differential geometry... a subject that is interseted in studying the "intrinsic geometry" of curves in euclidean space. Riemann discovered the intrinsic geometry of any smooth surface in euclidean space could be described with a set of coordinates that can be defined entirely within the surface and a (possibly) noneuclidean metric on those coordinates.

So we have come to an interesting discovery; we have presumed that space looks euclidean on small scales, which means that space can be described by a riemann manifold... however, riemann manifolds arose from the study of surfaces embedded in higher dimensional space! Becuase the two concepts have identical mathematical forms, all of the mathematics involved in the study of curves can be applied to the study of the universe.

Thus, when studying space, we use terms like "geodesics" and "curvature". These terms originally appeared when studying honest to goodness surfaces in euclidean spaces, but since the study of the universe has the same mathematical form as the study of surfaces, we use the same terms in our study of the universe.

General relativity was created using a very close relative of a riemann manifold; the construction instead presumes that the geometry space looks like the geometry of special relativity on small scales. Like with special relativity, Einstein imposed a single condition on this manifold and explored where the equations led. (The condition is that inertia and gravity are the same... presumed because the exact same quantity "mass" is used both in the classical description of inertia and the classical description of gravity and thought experiments that demonstrated the two concepts were sometimes indistinguishable, at least sometimes)

Even if Einstein's additional condition was incorrect, differential geometry is still the better way to describe the universe; our observations are necessarily small in scale, so we can only prove what the universe looks like on small scales. Of course, we're still presuming the small scale geometry looks the same everywhere, so differential manifolds aren't the perfect choice of study, but they're far more reasonable than the assumption that space looks euclidean as a whole. Even if space is euclidean as a whole, euclidean space is merely a special kind of differentiable manifold, we we would still have the correct description of space.

Originally posted by Hurkyl One of the things Astronomers get flak for is presuming that the universe is curved (or more precisely, their model permits the universe to curve). People think astronomers are losing touch by using these horribly abstract mathematical ideas that obviously can have no connection to physical reality!

Curved space is simply a mathematical tool for measuring density gradients (gravitational fields) in the superfluid "quantum vacuum". When you abstract it from this real substance the math does indeed get disconnected from reality. This error is where black-holes (singularities) and the Big Bang come from.

Originally posted by Hurkyl One of the things Astronomers get flak for is presuming that the universe is curved (or more precisely, their model permits the universe to curve). People think astronomers are losing touch by using these horribly abstract mathematical ideas that obviously can have no connection to physical reality!

But people don't know the reasons!

People think that astronomers are making all these additional assumptions about how the universe must look... but what they don't realize is that the exact opposite is true; astronomers are removing assumptions.

Something that people don't realize is that the "obviously correct" idea of a Euclidean space is chock full with a lot of assumptions; space looks the same on small and large scales (I mean the geometry, not necessarily the content)... squares exist... perfectly parallel lines exist... spheres have insides and outsides... lines intersect in at most one point...

All in all, when keeping with the spirit of Euclid, Hilbert axiomized euclidean geometry using fifteen postulates... and that's just for the geometry of the plane! People may remember that Euclid only had 5 postulates, but what they don't realize he also included some obvious rules which amount to additional assumptions, and Euclid was even incomplete! For example from Euclid's postulates you cannot prove that a line segment with one endpoint inside of a circle and one endpoint outside of a circle actually intersects the circle!

If we take a few steps back, we realize we're making a lot of assumptions, but then again, space certainly looks euclidean, doesn't it? But we realize that we're only looking at the space near us. It would make sense, then, presuming the universe is Euclidean on small scales is a much more reasonable law than presuming the entire universe is Euclidean!

And that's just what astronomers do; instead of presuming the entire universe is Euclidean, they presume it simply looks Euclidean on small scales (and may or may not look Euclidean globally). The mathematician Riemann studied geometries that look Euclidean on small scales, so today we call such things "Riemann manifolds". (they're a special type of "differentiable manifold")

However, as often happens in mathematics, the above reasoning is not why Riemann was studying his manifolds! He was active in a subject called differential geometry... a subject that is interseted in studying the "intrinsic geometry" of curves in euclidean space. Riemann discovered the intrinsic geometry of any smooth surface in euclidean space could be described with a set of coordinates that can be defined entirely within the surface and a (possibly) noneuclidean metric on those coordinates.

So we have come to an interesting discovery; we have presumed that space looks euclidean on small scales, which means that space can be described by a riemann manifold... however, riemann manifolds arose from the study of surfaces embedded in higher dimensional space! Becuase the two concepts have identical mathematical forms, all of the mathematics involved in the study of curves can be applied to the study of the universe.

Thus, when studying space, we use terms like "geodesics" and "curvature". These terms originally appeared when studying honest to goodness surfaces in euclidean spaces, but since the study of the universe has the same mathematical form as the study of surfaces, we use the same terms in our study of the universe.

General relativity was created using a very close relative of a riemann manifold; the construction instead presumes that the geometry space looks like the geometry of special relativity on small scales. Like with special relativity, Einstein imposed a single condition on this manifold and explored where the equations led. (The condition is that inertia and gravity are the same... presumed because the exact same quantity "mass" is used both in the classical description of inertia and the classical description of gravity and thought experiments that demonstrated the two concepts were sometimes indistinguishable, at least sometimes)

Even if Einstein's additional condition was incorrect, differential geometry is still the better way to describe the universe; our observations are necessarily small in scale, so we can only prove what the universe looks like on small scales. Of course, we're still presuming the small scale geometry looks the same everywhere, so differential manifolds aren't the perfect choice of study, but they're far more reasonable than the assumption that space looks euclidean as a whole. Even if space is euclidean as a whole, euclidean space is merely a special kind of differentiable manifold, we we would still have the correct description of space.

Hurkyl, so, you think the model of the universe's evolvement seen in my signature below is way off...?

I was thinking that the idea of a "saddle" shaped universe probably comes from observing a small part of this rather large spatter of condensed radiation and light we call the universe... or that we assume to be what the universe looks like.

I feel there will always be otherways to discribe the physical attributes of the universe. There is a cause for every effect. There are so many different effects in this universe and for each one there is a difference cause.

Each dimension creates its own effect... for example.

Not that I know... and I'm not assuming anything. But your expose has opened up more possiblities. thanks.

Yes and saddle shapes and spheres are meaningless without being concieved of in some euclidean meta-space.

See:

Thus, when studying space, we use terms like "geodesics" and "curvature". These terms originally appeared when studying honest to goodness surfaces in euclidean spaces, but since the study of the universe has the same mathematical form as the study of surfaces, we use the same terms in our study of the universe.

Originally posted by Hurkyl Thus, when studying space, we use terms like "geodesics" and "curvature". These terms originally appeared when studying honest to goodness surfaces in euclidean spaces, but since the study of the universe has the same mathematical form as the study of surfaces, we use the same terms in our study of the universe.

Unless of course the Universe is infinite then it does NOT have the same mathematical form as an "honest to goodness" surface. In fact it has no surface whatsoever. Such an extrapolation then becomes dishonest.

The study of infinite universes would be formally the same as the study of infinite surfaces.

The study of surfaces also includes studing surfaces of dimension higher than 2 in spaces of dimension higher than 3. In particular, it can be shown that the study of three dimensional universes is formally the same as the study of 3 dimensional surfaces in some space of dimension greater than or equal to 3.

Originally posted by Hurkyl The study of surfaces also includes studing surfaces of dimension higher than 2 in spaces of dimension higher than 3. In particular, it can be shown that the study of three dimensional universes is formally the same as the study of 3 dimensional surfaces in some space of dimension greater than or equal to 3.

And this is where mathematics escapes into it's own little fantasy universe devoid of any causal restictions.

How can something infinite in extent (in all directions) have a surface?

For total precision, I'm unsure if the appropriate definition would be infinite area or if it would be that there is no global bound on what the minimum distance between two points is. (of course, for an n-dimensional surface, I'm talking about n-dimensional area)

An example of an infinite surface is simply the Euclidean plane embedded in Euclidean 3-space. (Of course, the Euclidean plane embedded in the Euclidean plane would work as well)

And this is where mathematics escapes into it's own little fantasy universe devoid of any causal restictions.

It's an abstraction that aids the study of other things, be it 3-dimensional universes, partial differential equations with 7 variables, or data mining in databases with hundreds of entries per record.

Originally posted by Hurkyl
For total precision, I'm unsure if the appropriate definition would be infinite area

Which is only dealing with idealistic infinite extension in 2 dimensions.

or if it would be that there is no global bound on what the minimum distance between two points is.

What is the difference?

An example of an infinite surface is simply the Euclidean plane embedded in Euclidean 3-space.

Which is simply a mathematical ideal. It does not exist and cannot exist in reality, of course.

You still have not explained how a universe of infinite extent in ALL directions could have a surface.

It's an abstraction that aids the study of other things, be it 3-dimensional universes, partial differential equations with 7 variables, or data mining in databases with hundreds of entries per record.

Since that's a good thing, I'm curious as to what you meant.

Of course it is a good thing, but when extrapolating these abstractions as explanations of physical reality one must be careful to maintain contact with the key features of the physical process to be described.

In the case of the universe, it is omnidirectional extension itself that is key. Thus it is meaningless to treat the extension of the universe as a 2d plane or surface.

Originally posted by Hurkyl
I refer you to
_____________________
(of course, for an n-dimensional surface, I'm talking about n-dimensional area)
___________________

Well that is a completely abstract extrapolation and has no bearing on reality.

I never said that either.

Ok, so if the Universe is infinite in spatial extent then it does not have a surface and all your mathematical surface transformations and extrapolations do not relate to it whatsoever.

I said the study of infinite universes would be formally the same as the study of infinite surfaces.

Which is meaningless in the real world.

The definition of a n-dimensional manifold is that at every point on the manifold, it locally looks like Rn. In other words, at every point it has n independant directions.

Only when mapped in other abstract spaces. This still does not address reality.

Since by "omnidirectional" I presume you mean that at every point in the universe there are 3 independant directions

Your assumption is incorrect. I am saying that the omni-directional extension of the universe can be mapped as a cartesian 3d manifold, or it can be mapped as an isotropic vector matrix or whatever. The universe, itself has no dimension. Dimension is a mental tool for quantifying reality.

you are simply requiring that space be described by a 3 dimensional manifold.

Nope. I am requiring that space is not composed of dimension and is entirely independent of it.

Well that is a completely abstract extrapolation and has no bearing on reality.

Except in the ways the abstraction is used to explain reality.

Incidentally, n-dimensional area is directly applicable to "reality". Length is synonomous with 1-dimensional area. Volume is synonomous with 3-dimensional area.

Ok, so if the Universe is infinite in spatial extent then it does not have a surface and all your mathematical surface transformations and extrapolations do not relate to it whatsoever.

(a) I said that the study of infinite universes corresponsd formally to the study of infinite surfaces. That does not require that the universe have a surface.

(b) I think you are misunderstanding the formal correspondence; the universe IS a surface (in the language of the correspondence)

Which is meaningless in the real world.

Except that it allows us to apply the techniques of studying infinite surfaces to the study of infinite universes.

Only when mapped in other abstract spaces. This still does not address reality.

Do you not presume the universe looks locally like R3?

Your assumption is incorrect. I am saying that the omni-directional extension of the universe can be mapped as a cartesian 3d manifold, or it can be mapped as an isotropic vector matrix or whatever. The universe, itself has no dimension. Dimension is a mental tool for quantifying reality.

Since cartesian 3d manifolds ARE (three dimensional) surfaces, I fail to see what your objection was.

Originally posted by Hurkyl
Except in the ways the abstraction is used to explain reality.

Which has its limitations especially when dealing with real extension independent of dimension.

Incidentally, n-dimensional area is directly applicable to "reality". Length is synonomous with 1-dimensional area. Volume is synonomous with 3-dimensional area.

of course, but that is only 3 cartesian dimensions

(a) I said that the study of infinite universes corresponsd formally to the study of infinite surfaces. That does not require that the universe have a surface.

If it doesn't have a surface then how can you meaningfully study it using surfaces?

(b) I think you are misunderstanding the formal correspondence; the universe IS a surface (in the language of the correspondence)

And that language is inapplicable to physical extension independent of dimension.

Except that it allows us to apply the techniques of studying infinite surfaces to the study of infinite universes.

Which causes a whole mess of confusion and this is my point.

Do you not presume the universe looks locally like R3?

No. The universe doesn't look like anything else. That is your critical mistake. An infinite Universe cannot look like a finite object.

Since cartesian 3d manifolds ARE (three dimensional) surfaces, I fail to see what your objection was. [/B]

The point was obvious. You can map physical extension in MANY ways. The Universe is not composed of dimension and is entirely independent of it. The universe, itself has no dimension. Dimension is a mental tool for quantifying reality.

There is physical extension and there is metrical extension. Physical extension is independent of dimension, and metrical extension is composed of it.

(a) R3 is not finite.
(b) Classical mechanics presumes the universe is R3... and your posts on other threads indicate that you think Classical mechanics is correct.

Classical mechanics was NOT correct. It was obviously quite flawed, but the revolution in Physics did not fix the core error. It simply kludged the whole mess together to give us the core paradoxes of the Uncertainty relations and the wave-particle duality.

And all of science is a tool for describing the ways we quantify the universe. So?

So using this insight try and extricate yourself from this mess.

Anyways, I can observe the universe behaving as a three dimensional space.

No you can inaccurately describe its behavior using 3 dimensions. That is quite different.

Because the rest of us know what we mean, I simply say that the unvierse is three dimensional.

As long as you remember that the universe is not composed of mathematics and thus is not composed of dimension either then your mathematics can more accurately describe it.